Convex Underestimation of C2 Continuous Functions by Piecewise Quadratic Perturbation

Clifford A. Meyer and Christodoulos A. Floudas Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA Abstract This paper presents an efficient branch and bound approach to address the global optimization of constrained optimization problems with twice differentiable functions. A lower bound on the global minimum is determined via a convex nonlinear programming problem in which all nonconvex functions are substituted by their convex underestimators. This work refines the classical BB eigenvalue perturbation method for the convex underestimation of twice differentiable functions. New convex underestimators are proposed based on a smooth, piecewise quadratic, perturbation function. The parameters, coefficients of the quadratic terms in the perturbation function, are calculated using eigenvalue analysis techniques. Formulae defining the linear coefficients and the constants of the piecewise quadratic perturbations function are derived from continuity, smoothness and end point conditions. The piecewise quadratic form of the perturbation is far more flexible that the quadratic form employed in the classical BB methodology. This flexibility and improved bounds on the values lead to a vast improvement over the classical aBB method.